Should junior change to 2nd after senior dies?

Generally speaking, when John Doe, Jr. is given his name, it stays with him. He can change it later, but if he doesn't, he keeps that name. If his father (his namesake - the person after whom he was named) dies, John Doe, Jr.'s name remains the same. He is still his father's son, or, if you like, his namesake's son.